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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 10th 2010
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   Author: Mike Shulman
   Format: MarkdownAdded some discussion of other ways to define doubly-weak [[double categories]].

   Added some discussion of other ways to define doubly-weak double categories.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 27th 2010
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   Author: zskoda
   Format: MarkdownItexAt [[virtual double category]] (or fc-multicategory) there is a statement that virtual double category generalized weak double category. The same statement is kind of in Leinster book, the table after Example 5.1.1. Tom says that *most familiar 2d structures are degenerate fc-multicategories*. Then he gives a table with 3 rows and 3 columns. The top row is no degeneracy. The columns are *no horizontal composition*, *weak horizontal composition*, *strict horizontal composition* and entries at 11,12,13 are fc-multicategory, weak double category, strict double category. How can the case of weak horizontal composition be a special case of *no* horizontal composition ?? Horizontal composition is a structure, not a property. Am I missing something ?

   At virtual double category (or fc-multicategory) there is a statement that virtual double category generalized weak double category. The same statement is kind of in Leinster book, the table after Example 5.1.1. Tom says that most familiar 2d structures are degenerate fc-multicategories. Then he gives a table with 3 rows and 3 columns. The top row is no degeneracy. The columns are no horizontal composition, weak horizontal composition, strict horizontal composition and entries at 11,12,13 are fc-multicategory, weak double category, strict double category. How can the case of weak horizontal composition be a special case of no horizontal composition ?? Horizontal composition is a structure, not a property. Am I missing something ?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeSep 27th 2010
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   Author: Mike Shulman
   Format: MarkdownItexIn the same way that a [[monoidal category]] is a special case of a [[multicategory]], and that a category with products is a special case of a category. A virtual double category does not have a composition operation itself, but it has sufficient structure enabling the further existence of composites to become a property.

   In the same way that a monoidal category is a special case of a multicategory, and that a category with products is a special case of a category. A virtual double category does not have a composition operation itself, but it has sufficient structure enabling the further existence of composites to become a property.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeSep 27th 2010
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   Author: Mike Shulman
   Format: MarkdownItexThe characterization of composites by a universal property can be found in section 5 of [this paper](http://arxiv.org/abs/0907.2460). One day it should make its way into the nLab.

   The characterization of composites by a universal property can be found in section 5 of this paper. One day it should make its way into the nLab.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeSep 27th 2010
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   Author: zskoda
   Format: MarkdownItexThat is a great answer. Thanks.

   That is a great answer. Thanks.